AdS3/CFT2 Guide

This is a comprehensive guide to the $AdS_3/CFT_2$ correspondence as the cleanest low-dimensional laboratory for gauge/gravity duality. The subject sits at a rare intersection:

• 3D Einstein gravity is simple enough to compute with, yet it contains boundary degrees of freedom, black holes, and nontrivial topology.
• 2D conformal field theory is constrained enough to solve many questions exactly, and in special families it is solvable all the way to dynamical data (OPE coefficients, higher-genus partition functions, etc.).

A key meta-lesson that will recur throughout this guide:

In $AdS_3/CFT_2$, symmetry + topology + modularity often replaces the role played by local bulk dynamics in higher-dimensional holography.

The goals here are:

1. Build a working toolkit: classical $AdS_3$ gravity, Brown–Henneaux boundary conditions, BTZ black holes, the Virasoro algebra, modular invariance, holographic renormalization, and the basic dictionary.
2. Bridge to current research: Virasoro blocks and semiclassical dynamics/chaos, higher-spin $AdS_3$, strings on $AdS_3$ (NS–NS WZW vs RR flux), integrability, solvable deformations like $T\bar T$, and the status of “pure” $AdS_3$ quantum gravity.

This is written for graduate students who already know:

• QFT basics (path integrals, symmetries, renormalization),
• GR basics (Einstein equations, horizons, thermodynamics),
• and a first pass of AdS/CFT in higher dimensions.

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Table of contents

• 0. Big picture and roadmap
• 1. Geometry of $AdS_3$
• 2. 3D Einstein gravity and boundary conditions
• 3. Brown–Henneaux and the emergence of Virasoro
• 4. BTZ black holes
• 5. CFT2 essentials: Virasoro, states, OPE, modular invariance
• 6. The Cardy formula and black hole entropy
• 7. The holographic dictionary in $AdS_3/CFT_2$
• 8. Correlators, Virasoro blocks, and semiclassical gravity
• 9. Entanglement in $AdS_3/CFT_2$
• 10. $AdS_3$ gravity as Chern–Simons theory
• 11. Higher-spin $AdS_3/CFT_2$ and $W$-algebras
• 12. Strings on $AdS_3$: NS–NS WZW, RR flux, D1–D5, and tensionless limits
• 13. Integrability in $AdS_3/CFT_2$
• 14. Solvable irrelevant deformations: $T\bar T$ and friends
• 15. “Pure” $AdS_3$ gravity, modular bootstrap, and the sum over saddles
• 16. Exercises and mini-projects
• 17. Annotated reading list

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0. Big picture and roadmap

0.1 What makes $AdS_3/CFT_2$ special?

• No local gravitons in 3D Einstein gravity, yet there are:

  • boundary gravitons (large diffeomorphisms that become physical at infinity),
  • black holes (BTZ),
  • and nontrivial global/topological physics (quotients, wormholes, etc.).

• Two-dimensional CFTs are heavily constrained:

  • by Virasoro symmetry,
  • modular invariance on the torus,
  • crossing symmetry of four-point functions,
  • and (in special models) extended chiral algebras and supersymmetry.

This means many holographic statements can be made precisely, often more sharply than in $AdS_5/CFT_4$.

0.2 The “three layers” of $AdS_3/CFT_2$

It helps to separate three increasingly strong notions of holography:

1. Universal semiclassical gravity layer
   Large central charge $c\gg 1$ and a sparse light spectrum: you recover BTZ thermodynamics, RT/HRT entanglement, and universal features of heavy-light correlators.

2. Topological quantum gravity layer
   3D gravity as (roughly) Chern–Simons theory: holonomies, Wilson lines, boundary WZW structures, edge modes, and an unusually explicit handle on what “degrees of freedom” mean.

3. Full string theory layer
   $AdS_3\times S^3\times M_4$ (with $M_4=T^4$ or K3) and the D1–D5 CFT: exactly solvable points (especially with NS–NS flux) and integrability-based control for RR/mixed flux.

This guide develops all three, and also highlights where they do not seamlessly match (this matters for understanding what “the” dual of “pure” $AdS_3$ gravity could mean).

0.3 Conventions and quick dictionary (read once; return often)

Boundary geometry. Unless stated otherwise, the boundary is a cylinder with coordinates $(t,\phi)$ and

• $\phi\sim \phi+2\pi$,
• Minkowski boundary metric $ds^2_\partial\sim -dt^2 + d\phi^2$,
• lightcone coordinates $x^\pm = t\pm \phi$,
• Euclidean time $t_E = i t$.

Central charge and bulk coupling.

$$c = \frac{3\ell}{2G},\qquad k=\frac{\ell}{4G},\qquad c=6k,$$

where $k$ is the Chern–Simons level in the $sl(2)\oplus sl(2)$ formulation.

CFT Hamiltonian on the circle. With spatial circle length $2\pi$,

$$H = L_0+\bar L_0 - \frac{c}{12},\qquad P = L_0-\bar L_0,$$

and energies are measured in units where the circumference is $2\pi$.

BTZ/CFT matching (rotating case).

$$L_0-\frac{c}{24}=\frac{\ell M+J}{2},\qquad \bar L_0-\frac{c}{24}=\frac{\ell M-J}{2}.$$

Equivalently, in terms of left/right temperatures,

$$L_0-\frac{c}{24}=\frac{\pi^2 c}{6}T_L^2,\qquad \bar L_0-\frac{c}{24}=\frac{\pi^2 c}{6}T_R^2,$$

and in terms of horizons,

$$T_L=\frac{r_+ + r_-}{2\pi \ell^2},\qquad T_R=\frac{r_+ - r_-}{2\pi \ell^2}.$$

Stress tensor normalization (flat boundary). In Fefferman–Graham gauge,

$$\langle T_{ij}\rangle = \frac{\ell}{8\pi G}\left(g^{(2)}_{ij}-g^{(0)}_{ij}\,\mathrm{Tr}\,g^{(2)}\right),$$

and the trace anomaly is

$$\langle T^i_{\ i}\rangle = \frac{c}{24\pi}R[g^{(0)}].$$

Bulk masses vs boundary weights. A bulk field with scaling dimension $\Delta$ corresponds to a CFT primary of weights $(h,\bar h)$ with

$$\Delta=h+\bar h,\qquad s=h-\bar h.$$

For a scalar of mass $m$,

$$\Delta(\Delta-2)=m^2\ell^2,\qquad \Delta = 1 \pm \sqrt{1+m^2\ell^2}.$$

0.4 What you should compute at least once

If you want a reliable internal map of the subject, do these “checkpoints” explicitly:

1. Derive the Brown–Henneaux central charge from holographic renormalization (or the canonical charge algebra).
2. Show that the general Brown–Henneaux solution is the Bañados metric labeled by $L(x^+),\bar L(x^-)$.
3. Derive BTZ entropy from Cardy using $c=3\ell/(2G)$.
4. Derive the RT formula for a single interval from a geodesic length, and match to a twist-operator computation in CFT.
5. Compute a heavy-light Virasoro vacuum block and see thermality emerge.
6. Translate BTZ smoothness into a holonomy condition in Chern–Simons variables.

The exercises in Section 16 are chosen to guide you through these.

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1. Geometry of $AdS_3$

1.1 Definition and embeddings

$AdS_3$ is a maximally symmetric Lorentzian manifold with constant negative curvature. A convenient definition is as the hyperboloid in $\mathbb{R}^{2,2}$:

$$-X_0^2 - X_3^2 + X_1^2 + X_2^2 = -\ell^2,$$

with ambient metric $ds^2 = -dX_0^2 - dX_3^2 + dX_1^2 + dX_2^2$.

The curvature is

$$R_{\mu\nu} = -\frac{2}{\ell^2} g_{\mu\nu},\qquad R = -\frac{6}{\ell^2}.$$

1.2 Global coordinates and conformal boundary

A standard global metric is

$$ds^2 = \ell^2\left(-\cosh^2\rho\, dt^2 + d\rho^2 + \sinh^2\rho\, d\phi^2\right),$$

with $\phi\sim \phi+2\pi$.

The conformal boundary is at $\rho\to\infty$. After the Weyl rescaling $ds^2\to e^{-2\rho}ds^2$, the induced boundary metric becomes that of a cylinder:

$$ds^2_{\partial} \sim -dt^2 + d\phi^2.$$

1.3 Poincaré patch

The Poincaré patch metric is

$$ds^2 = \frac{\ell^2}{z^2}\left(dz^2 - dt^2 + dx^2\right),\qquad z>0,$$

with boundary at $z\to 0$. This patch is convenient for local operator insertions and for the simplest bulk-to-boundary propagators.

1.4 Isometries and the two copies of $SL(2,\mathbb{R})$

The isometry group is $SO(2,2)$, which factorizes as

$$SO(2,2)\cong \frac{SL(2,\mathbb{R})_L\times SL(2,\mathbb{R})_R}{\mathbb{Z}_2}.$$

This factorization foreshadows the two Virasoro algebras in the boundary CFT, and it is also the reason $AdS_3$ can be described naturally in terms of two $sl(2,\mathbb{R})$ Chern–Simons connections.

1.5 Euclidean $AdS_3$ and hyperbolic geometry

After Wick rotation $t\to -i t_E$, Euclidean $AdS_3$ is hyperbolic space $H^3$. This viewpoint matters because:

• Euclidean bulk saddle points are often hyperbolic 3-manifolds with prescribed conformal boundary (a Riemann surface),
• and many questions about “the sum over saddles” become questions in hyperbolic geometry and Teichmüller theory.

For example, the Euclidean continuation of thermal states makes the boundary a torus, and different bulk saddles correspond to different choices of the contractible cycle in the solid torus.

1.6 Quotients, conjugacy classes, and “what BTZ really is”

Many physically relevant solutions are discrete quotients:

$$AdS_3/\Gamma,$$

with $\Gamma\subset SO(2,2)$ (or $SL(2,\mathbb{R})\times SL(2,\mathbb{R})$).

On the Euclidean side, elements in $SL(2,\mathbb{C})$ are classified as elliptic/parabolic/hyperbolic. In Lorentzian $SL(2,\mathbb{R})$, the analogous classification governs:

• conical defects (elliptic holonomy),
• massless BTZ (parabolic),
• BTZ black holes (hyperbolic).

This is the cleanest “group-theoretic” way to remember the phase diagram of classical solutions.

1.7 Boundary-anchored geodesics: the workhorse for RT and correlators

In $AdS_3$, boundary-anchored spacelike geodesics are ubiquitous because:

• RT surfaces for single intervals are geodesics,
• geodesic Witten diagrams approximate heavy exchanges,
• and geodesic lengths encode two-point functions in the large-$\Delta$ limit.

A representative formula (global AdS, equal-time interval of angular size $\Delta\phi$) is

$$\frac{\text{Length}}{\ell} = 2\log\left(\frac{2}{\epsilon}\sin\frac{\Delta\phi}{2}\right),$$

where $\epsilon$ is a UV cutoff near the boundary. Plugging this into

$$S_{\text{EE}}=\frac{\text{Length}}{4G}$$

gives the universal CFT result

$$S_{\text{EE}} = \frac{c}{3}\log\left(\frac{2}{\epsilon}\sin\frac{\Delta\phi}{2}\right),$$

using $c=3\ell/(2G)$.

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2. 3D Einstein gravity and boundary conditions

2.1 Action and variational principle

The Einstein–Hilbert action with negative cosmological constant is

$$S_{\text{bulk}} = \frac{1}{16\pi G}\int_{\mathcal{M}} d^3x\,\sqrt{-g}\left(R + \frac{2}{\ell^2}\right).$$

A well-posed Dirichlet variational principle requires the Gibbons–Hawking term

$$S_{\text{GH}} = \frac{1}{8\pi G}\int_{\partial\mathcal{M}} d^2x\,\sqrt{-\gamma}\,K,$$

plus local counterterms $S_{\text{ct}}$ to render the on-shell action finite.

In $AdS_3$ a minimal counterterm is

$$S_{\text{ct}} = -\frac{1}{8\pi G}\int_{\partial\mathcal{M}} d^2x\,\sqrt{-\gamma}\,\frac{1}{\ell}.$$

2.2 Fefferman–Graham expansion in 3D

Near the boundary one can use Fefferman–Graham (FG) gauge:

$$ds^2 = \ell^2\frac{dz^2}{z^2} + \frac{1}{z^2} g_{ij}(x,z)\,dx^i dx^j.$$

In 3D Einstein gravity, the expansion truncates:

$$g_{ij}(x,z) = g^{(0)}_{ij}(x) + z^2 g^{(2)}_{ij}(x) + z^4 g^{(4)}_{ij}(x),$$

with

$$g^{(4)}_{ij} = \frac{1}{4} g^{(2)}_{ik}\,g^{(0)kl}\,g^{(2)}_{lj}.$$

The Einstein equations impose constraints on $g^{(2)}$:

$$\mathrm{Tr}\,g^{(2)} \equiv g^{(0)ij}g^{(2)}_{ij} = -\frac{1}{2}R[g^{(0)}],$$

and

$$\nabla^{(0)i} g^{(2)}_{ij} = \nabla^{(0)}_j(\mathrm{Tr}\,g^{(2)}),$$

where $\nabla^{(0)}$ is the covariant derivative for $g^{(0)}$.

For a flat boundary metric $g^{(0)}_{ij}=\eta_{ij}$, these reduce to:

• $g^{(2)}$ is traceless,
• and $g^{(2)}$ is conserved.

This is the bulk origin of the CFT stress tensor Ward identities.

2.3 Holographic (Brown–York) stress tensor

The renormalized boundary stress tensor is

$$T_{ij} = \frac{2}{\sqrt{-g^{(0)}}}\frac{\delta S_{\text{ren}}}{\delta g^{(0)ij}} = \frac{1}{8\pi G}\left(K_{ij}-K\gamma_{ij}-\frac{1}{\ell}\gamma_{ij}\right)\Big|_{\text{ren}},$$

and in FG gauge it becomes

$$\langle T_{ij}\rangle = \frac{\ell}{8\pi G}\left(g^{(2)}_{ij}-g^{(0)}_{ij}\,\mathrm{Tr}\,g^{(2)}\right).$$

Taking the trace gives the 2D Weyl anomaly:

$$\langle T^i_{\ i}\rangle = \frac{\ell}{16\pi G}R[g^{(0)}] = \frac{c}{24\pi}R[g^{(0)}],$$

hence

$$c = \frac{3\ell}{2G}.$$

2.4 Boundary conditions beyond Brown–Henneaux (optional but important)

Brown–Henneaux boundary conditions fix the boundary metric up to a Weyl factor and yield two Virasoro algebras. There are many other consistent boundary conditions in 3D gravity (chiral boundary conditions, “soft hair” boundary conditions, warped boundary conditions, etc.) leading to:

• Virasoro $\times$ Kac–Moody,
• BMS$_3$-like algebras in flat-space limits,
• or different state spaces.

For this guide we mostly stick to Brown–Henneaux because it is the standard setting of $AdS_3/CFT_2$.

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3. Brown–Henneaux and the emergence of Virasoro

3.1 Brown–Henneaux falloffs and asymptotic symmetries

A standard form of Brown–Henneaux boundary conditions (in coordinates where $r\to\infty$ is the boundary) is:

$$g_{tt} = -\frac{r^2}{\ell^2}+O(1),\quad g_{\phi\phi} = r^2+O(1),\quad g_{t\phi}=O(1),$$ $$g_{rr} = \frac{\ell^2}{r^2}+O(r^{-4}),\quad g_{rt}=O(r^{-3}),\quad g_{r\phi}=O(r^{-3}).$$

The diffeomorphisms preserving these falloffs are parametrized by two arbitrary functions of one variable,

$$\epsilon^+(x^+),\qquad \epsilon^-(x^-),\qquad x^\pm=t\pm \phi,$$

and their charge algebra yields:

• two copies of the Witt algebra at the classical level,
• centrally extended to Virasoro with $$c=\frac{3\ell}{2G}.$$

Conceptual point: the central charge is not “put in by hand”; it is the price paid by the boundary terms in the gravitational charge algebra.

3.2 Boundary gravitons as Virasoro descendants

Since there are no propagating gravitons in 3D, the excitations around global $AdS_3$ are boundary gravitons, created by acting with Virasoro generators on the vacuum:

$$L_{-n_1}L_{-n_2}\cdots |0\rangle,\qquad n_i\ge 2,$$

(and similarly for $\bar L_{-n}$). The $n=1$ modes correspond to global $SL(2,\mathbb{R})$ isometries and do not generate independent physical states in the vacuum module.

The one-loop partition function around thermal $AdS_3$ is the vacuum character (up to the classical piece):

$$Z_{\text{1-loop}}(\tau,\bar\tau) \sim |q|^{-c/12}\prod_{n=2}^\infty \frac{1}{|1-q^n|^2},\qquad q=e^{2\pi i \tau}.$$

3.3 The Bañados geometries: the general Brown–Henneaux solution

A major simplification in 3D: every classical solution of Einstein gravity with Brown–Henneaux boundary conditions is locally $AdS_3$, and the global data can be packaged into two functions.

In FG-like coordinates $(\rho,x^+,x^-)$, the general solution can be written as the Bañados metric:

$$ds^2 = \ell^2 d\rho^2 + \left(e^{2\rho} + \frac{1}{4}e^{-2\rho}L(x^+)\bar L(x^-)\right)dx^+dx^- + L(x^+)dx^{+2} + \bar L(x^-)dx^{-2}.$$

Here $L(x^+)$ and $\bar L(x^-)$ are arbitrary functions (subject to periodicity on the cylinder). They are the classical counterparts of the boundary stress tensor expectation values.

For a flat boundary metric,

$$\langle T_{++}(x^+)\rangle = \frac{\ell}{8\pi G}\,L(x^+)=\frac{c}{12\pi}L(x^+), \qquad \langle T_{--}(x^-)\rangle = \frac{\ell}{8\pi G}\,\bar L(x^-)=\frac{c}{12\pi}\bar L(x^-).$$

Expanding in Fourier modes,

$$L(x^+) = \sum_{n} L_n e^{-inx^+},\qquad \bar L(x^-) = \sum_n \bar L_n e^{-inx^-},$$

the $L_n,\bar L_n$ become (after quantization) the Virasoro generators.

This is the most concrete realization of the slogan: “boundary gravitons are Virasoro descendants.”

3.4 Virasoro coadjoint orbits and semiclassical states (optional but powerful)

The functions $L(x^+)$ label coadjoint orbits of the Virasoro group. From the CFT viewpoint:

• a classical background $L(x^+)$ corresponds to a stress tensor expectation value in some state,
• and large diffeomorphisms act by the Schwarzian transformation law of $T$.

This viewpoint is central in modern work connecting:

• 3D gravity path integrals on fixed topology,
• Teichmüller theory,
• and the appearance of Virasoro conformal blocks as building blocks of gravitational amplitudes.

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4. BTZ black holes

4.1 Metric and parameters

The BTZ black hole is a quotient of $AdS_3$. A convenient form of the rotating BTZ metric is

$$ds^2 = -\frac{(r^2-r_+^2)(r^2-r_-^2)}{\ell^2 r^2}\,dt^2 + \frac{\ell^2 r^2}{(r^2-r_+^2)(r^2-r_-^2)}\,dr^2 + r^2\left(d\phi - \frac{r_+r_-}{\ell r^2}dt\right)^2,$$

with $\phi\sim \phi+2\pi$.

The ADM mass $M$ and angular momentum $J$ are

$$M=\frac{r_+^2+r_-^2}{8G\ell^2},\qquad J=\frac{r_+r_-}{4G\ell}.$$

The horizon circumference is $2\pi r_+$, and the Bekenstein–Hawking entropy is

$$S_{\text{BH}} = \frac{\text{Area}}{4G} = \frac{2\pi r_+}{4G}.$$

Useful special cases.

• Global $AdS_3$ corresponds to $M=-\frac{1}{8G}$ and $J=0$.
• Conical defects correspond to $-\frac{1}{8G}<M<0$ (elliptic holonomy).
• The massless BTZ geometry has $M=J=0$ (parabolic holonomy).
• BTZ black holes have $M>0$ and satisfy $|J|\le \ell M$ (hyperbolic holonomy).
• Extremal BTZ is $|J|=\ell M$ (so $r_+=|r_-|$).

These align cleanly with the CFT threshold at $h,\bar h\sim c/24$.

4.2 Thermodynamics and left/right movers

The Hawking temperature and angular velocity are

$$T_H=\frac{r_+^2-r_-^2}{2\pi \ell^2 r_+},\qquad \Omega_H=\frac{r_-}{\ell r_+}.$$

Define left- and right-moving temperatures

$$T_L=\frac{r_+ + r_-}{2\pi \ell^2},\qquad T_R=\frac{r_+ - r_-}{2\pi \ell^2},$$

so that

$$T_H = \frac{2T_LT_R}{T_L+T_R},\qquad \Omega_H = \frac{T_L-T_R}{T_L+T_R}.$$

From the CFT point of view, the rotating thermal ensemble is naturally written in terms of $L_0,\bar L_0$:

$$Z(\beta_L,\beta_R) = \mathrm{Tr}\left(e^{-\beta_L\left(L_0-\frac{c}{24}\right)}e^{-\beta_R\left(\bar L_0-\frac{c}{24}\right)}\right),$$

with $\beta_{L,R}=1/T_{L,R}$.

4.3 Euclidean BTZ, boundary torus, and modular transformations

Euclidean continuation makes the boundary geometry a torus. The BTZ saddle is associated to the choice of which cycle on the boundary torus becomes contractible in the bulk solid torus.

Thermal $AdS_3$ corresponds to the “other” choice. Exchanging these saddles is a modular $S$ transformation of the boundary torus, and this bulk Hawking–Page transition is the geometric avatar of the CFT modular transformation that drives the Cardy formula.

A very useful boundary description uses $x^\pm$ coordinates and identifies

$$x^+ \sim x^+ + i\beta_L,\qquad x^- \sim x^- + i\beta_R,$$

so that the complex modular parameters can be taken as

$$\tau = i\frac{\beta_L}{2\pi},\qquad \bar\tau = -i\frac{\beta_R}{2\pi}$$

(on a spatial circle of length $2\pi$).

4.4 BTZ as a constant-stress-tensor (Bañados) geometry

BTZ is a special case of the Bañados family with constant

$$L(x^+) = L_0,\qquad \bar L(x^-)=\bar L_0.$$

Depending on the values of $L_0,\bar L_0$ (relative to the $AdS_3$ vacuum), the geometry is:

• a conical defect,
• a massless BTZ,
• or a BTZ black hole.

From the CFT viewpoint, this is precisely the statement that heavy states are characterized at leading order by constant stress tensor expectation values.

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5. CFT2 essentials: Virasoro, states, OPE, modular invariance

This section is a “CFT2 crash course” tuned to what you actually use in $AdS_3/CFT_2$.

5.1 Virasoro algebra and the stress tensor OPE

In complex coordinates $(z,\bar z)$ on the plane, the conformal symmetry enhances to the Virasoro algebra:

$$[L_m,L_n] = (m-n)L_{m+n} + \frac{c}{12}m(m^2-1)\delta_{m+n,0},$$

and similarly for $\bar L_n$.

The stress tensor OPE encodes this algebra:

$$T(z)T(0) = \frac{c/2}{z^4} + \frac{2T(0)}{z^2} + \frac{\partial T(0)}{z} + \cdots.$$

5.2 Primaries, descendants, and state–operator correspondence

A primary operator $\mathcal{O}(z,\bar z)$ has weights $(h,\bar h)$ if

$$[L_0,\mathcal{O}(0)] = h\,\mathcal{O}(0),\qquad [L_{n>0},\mathcal{O}(0)]=0,$$

and similarly for $\bar L_n$.

Descendants are obtained by acting with $L_{-n}$, $\bar L_{-n}$ with $n>0$.

The scaling dimension and spin are

$$\Delta = h+\bar h,\qquad s=h-\bar h.$$

Under the state–operator map, a primary corresponds to a highest-weight state:

$$|\mathcal{O}\rangle = \mathcal{O}(0)|0\rangle, \qquad L_0|\mathcal{O}\rangle = h|\mathcal{O}\rangle,\quad L_{n>0}|\mathcal{O}\rangle = 0,$$

and similarly for the right-moving sector.

5.3 Ward identities and the Schwarzian transformation law

The stress tensor insertion obeys the Ward identity

$$\left\langle T(z)\prod_i \mathcal{O}_i(z_i)\right\rangle = \sum_i\left(\frac{h_i}{(z-z_i)^2}+\frac{1}{z-z_i}\partial_{z_i}\right) \left\langle \prod_i \mathcal{O}_i(z_i)\right\rangle.$$

Under a conformal map $z=z(w)$, the stress tensor transforms as

$$T(w) = \left(\frac{dz}{dw}\right)^2 T(z) + \frac{c}{12}\{z,w\},$$

where the Schwarzian derivative is

$$\{z,w\} \equiv \frac{z'''(w)}{z'(w)} - \frac{3}{2}\left(\frac{z''(w)}{z'(w)}\right)^2.$$

This formula is the CFT side of many bulk statements about “large diffeomorphisms,” and it is also the engine behind heavy-light thermality (a heavy operator sources $T$, and a coordinate change can absorb it).

5.4 Operator product expansion and conformal blocks

Four-point functions decompose into conformal blocks:

$$\langle \mathcal{O}_1(0)\mathcal{O}_2(z)\mathcal{O}_3(1)\mathcal{O}_4(\infty)\rangle = \sum_p C_{12p}C_{34p}\,\mathcal{F}_p(z)\,\overline{\mathcal{F}}_p(\bar z),$$

where $\mathcal{F}_p(z)$ are Virasoro conformal blocks (or blocks of an extended chiral algebra).

In $AdS_3/CFT_2$, you should keep the following correspondences in mind:

• Virasoro vacuum block $\leftrightarrow$ semiclassical “pure gravity” contribution (gravitons/boundary gravitons).
• Exchange of a primary $\leftrightarrow$ exchange of a bulk particle/field.
• Heavy primaries with $h,\bar h\sim O(c)$ $\leftrightarrow$ classical geometries (conical defects or BTZ).
• Crossing symmetry $\leftrightarrow$ bulk factorization/consistency.

5.5 Torus partition function and modular invariance

The torus partition function is

$$Z(\tau,\bar\tau)=\mathrm{Tr}\left(q^{L_0-\frac{c}{24}}\bar q^{\bar L_0-\frac{c}{24}}\right),\qquad q=e^{2\pi i\tau}.$$

Consistency requires invariance under modular transformations:

$$\tau\to \frac{a\tau+b}{c\tau+d},\qquad \begin{pmatrix}a&b\\c&d\end{pmatrix}\in SL(2,\mathbb{Z}).$$

The $S$ transformation $\tau\to -1/\tau$ is the engine behind the Cardy formula and, on the bulk side, behind the exchange between thermal $AdS_3$ and BTZ saddles.

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6. The Cardy formula and black hole entropy

6.1 A derivation you should know (modular invariance $\Rightarrow$ Cardy)

On a circle of length $2\pi$, the Euclidean thermal partition function with inverse temperature $\beta$ is

$$Z(\beta)=\mathrm{Tr}\left(e^{-\beta (L_0+\bar L_0-\frac{c}{12})}\right).$$

Relate this to the torus parameter via $\tau=i\beta/(2\pi)$ (non-rotating case). Modular $S$ gives

$$Z(\beta) = Z\!\left(\frac{4\pi^2}{\beta}\right).$$

At high temperature ($\beta\ll 1$), $Z(\beta)$ is controlled by low temperature ($4\pi^2/\beta\gg 1$), where the vacuum dominates:

$$Z\!\left(\frac{4\pi^2}{\beta}\right) \approx \exp\left(\frac{\pi^2 c}{3\beta}\right),$$

hence

$$\log Z(\beta) \approx \frac{\pi^2 c}{3\beta}\qquad (\beta\ll 1).$$

A saddle-point inverse Laplace transform then yields the Cardy density of states.

6.2 Cardy growth of states (left/right form)

For large left/right energies, Cardy gives

$$S_{\text{Cardy}} \approx 2\pi\sqrt{\frac{c}{6}\left(L_0-\frac{c}{24}\right)} + 2\pi\sqrt{\frac{c}{6}\left(\bar L_0-\frac{c}{24}\right)}.$$

More refined versions include:

• an effective central charge $c_{\mathrm{eff}}$ if the lowest weight is not zero,
• Rademacher expansions (exact modular sums) for subleading corrections,
• and logarithmic corrections that match one-loop bulk determinants.

6.3 Matching BTZ entropy

Using the Brown–Henneaux relation $c=3\ell/(2G)$, the BTZ charges match CFT weights via

$$L_0-\frac{c}{24}=\frac{\ell M+J}{2},\qquad \bar L_0-\frac{c}{24}=\frac{\ell M-J}{2}.$$

In terms of $r_\pm$ this becomes

$$L_0-\frac{c}{24}=\frac{(r_+ + r_-)^2}{16G\ell},\qquad \bar L_0-\frac{c}{24}=\frac{(r_+ - r_-)^2}{16G\ell}.$$

Plugging into Cardy gives

$$S_{\text{Cardy}} = 2\pi\frac{r_+ + r_-}{8G} + 2\pi\frac{r_+ - r_-}{8G} = \frac{2\pi r_+}{4G} = S_{\text{BH}}.$$

This is the classic “precision check” of $AdS_3/CFT_2$.

6.4 Physical interpretation: the black hole threshold at $\Delta\sim c$

In holographic CFTs, there is a sharp threshold around

$$h,\bar h \sim \frac{c}{24}.$$

• Below threshold: states are described semiclassically by particles/conical defects plus boundary gravitons.
• Above threshold: typical states behave thermodynamically like BTZ black holes.

This organizing principle is a cornerstone of the “universal semiclassical layer” of $AdS_3/CFT_2$.

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7. The holographic dictionary in $AdS_3/CFT_2$

7.1 Parameter matching

The most universal relation is

$$c=\frac{3\ell}{2G}.$$

In “pure” Einstein gravity, this essentially fixes the bulk coupling in terms of the boundary central charge.

In string embeddings one also has additional parameters:

• string length $\ell_s$ (via $\alpha'$),
• flux integers (e.g. NS–NS level $k$),
• internal manifold data ($T^4$, K3),
• and CFT data such as the D1–D5 charges.

7.2 Boundary sources vs bulk boundary conditions

A clean operational statement of the dictionary is:

• Choose boundary data (sources) for CFT operators.
• Impose corresponding boundary conditions for bulk fields.
• The renormalized on-shell bulk action is the generating functional of CFT correlators.

For example, the boundary metric $g^{(0)}$ sources the stress tensor, and the FG coefficient $g^{(2)}$ encodes $\langle T\rangle$.

7.3 Fields and operators: masses, dimensions, and spins

A scalar field of mass $m$ corresponds to a primary operator with dimension

$$\Delta = 1 + \sqrt{1+m^2\ell^2},$$

or alternatively $\Delta = 1 - \sqrt{1+m^2\ell^2}$ for alternate quantization when allowed ($-1<m^2\ell^2<0$).

For spin-$s$ fields, it is convenient to package data as left/right weights:

$$\Delta=h+\bar h,\qquad s=h-\bar h.$$

In particular:

• conserved currents correspond to massless gauge fields,
• and the stress tensor corresponds to the bulk graviton.

7.4 States and geometries (a practical map)

Canonical correspondences:

• CFT vacuum $\leftrightarrow$ global $AdS_3$.
• Thermal state on a circle $\leftrightarrow$ Euclidean BTZ saddle (at high temperature).
• Heavy primary state $\leftrightarrow$ classical quotient geometry (conical defect or BTZ, depending on $h,\bar h$).
• Virasoro descendants $\leftrightarrow$ boundary graviton excitations (Bañados with nontrivial $L(x^+),\bar L(x^-)$).

7.5 Thermodynamics as modular invariance

The boundary torus partition function encodes bulk saddle dominance:

• Low temperature: thermal $AdS_3$ dominates.
• High temperature: BTZ dominates.

In the bulk, this is Hawking–Page; in the boundary, it is modular invariance. You should think of these as the same statement viewed from two sides.

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8. Correlators, Virasoro blocks, and semiclassical gravity

This section is where $AdS_3/CFT_2$ becomes “computationally alive.”

8.1 Two-point functions and the geodesic approximation

For a scalar primary $\mathcal{O}$ with large dimension $\Delta\gg 1$, the boundary two-point function is dominated by a bulk geodesic:

$$\langle \mathcal{O}(x)\mathcal{O}(y)\rangle \sim e^{-\Delta\,\text{Length}(x,y)/\ell}.$$

This connects the semiclassical CFT limit to the simplest bulk computation and is the entry point for RT (replace $\Delta$ by the twist dimension).

8.2 Conformal blocks as bulk “exchange channels”

For four-point functions, conformal blocks implement factorization. In large-$c$ holographic CFTs:

• Global blocks correspond roughly to exchanging a single bulk field in a fixed background.
• Virasoro blocks resum an infinite tower of multi-stress-tensor exchanges, i.e. “gravitational dressing.”

The most important block is the Virasoro vacuum block, which captures boundary graviton exchange and controls many universal late-time/thermal features.

8.3 Heavy–light limit and emergent thermality

Consider heavy operators $\mathcal{O}_H$ with dimensions scaling as $h_H,\bar h_H\sim O(c)$, and light operators $\mathcal{O}_L$ with $h_L,\bar h_L\ll c$.

A standard setup is

$$\langle \mathcal{O}_H(\infty)\mathcal{O}_H(0)\mathcal{O}_L(1)\mathcal{O}_L(z)\rangle.$$

At large $c$, the vacuum Virasoro block in this correlator often takes a remarkably simple form after a uniformizing coordinate change. One convenient parameterization is

$$\alpha \equiv \sqrt{1-\frac{24h_H}{c}}.$$

• If $h_H < c/24$, then $\alpha$ is real and the heavy state looks like a conical defect.
• If $h_H > c/24$, then $\alpha = i\lambda$ with $\lambda=\sqrt{\frac{24h_H}{c}-1}$, and the heavy state behaves thermally with an effective temperature $$\beta_{\mathrm{eff}} = \frac{2\pi}{\lambda},\qquad T_{\mathrm{eff}}=\frac{\lambda}{2\pi}.$$

This is one of the sharpest demonstrations that black hole thermality is encoded in Virasoro symmetry in the large-$c$ regime.

8.4 Finite-temperature correlators (and BTZ) from conformal maps

On the Euclidean cylinder $w=\sigma+i\tau$ with $\tau\sim\tau+\beta$, map to the plane via

$$z = e^{\frac{2\pi}{\beta}w}.$$

For a primary of weights $(h,\bar h)$, the thermal two-point function becomes

$$\langle \mathcal{O}(w,\bar w)\mathcal{O}(0,0)\rangle_\beta = \left(\frac{\pi/\beta}{\sin\left(\frac{\pi}{\beta}w\right)}\right)^{2h} \left(\frac{\pi/\beta}{\sin\left(\frac{\pi}{\beta}\bar w\right)}\right)^{2\bar h}.$$

Analytically continuing $\tau\to it$ yields the real-time thermal correlator. For spinless operators ($h=\bar h=\Delta/2$) one obtains the familiar $\sinh$ form.

From the bulk side, this correlator is the boundary limit of a bulk propagator in Euclidean BTZ (solid torus). This is a direct “correlator-level” check of the dictionary.

8.5 Out-of-time-order correlators and maximal chaos (what to know)

One of the striking applications of Virasoro vacuum blocks is to out-of-time-order correlators (OTOCs), which diagnose quantum chaos.

A schematic OTOC is

$$\langle W(t)V(0)W(t)V(0)\rangle_\beta,$$

with an analytic continuation that sends the cross-ratio $z$ around a branch cut.

In large-$c$ CFTs with a sparse spectrum, the Virasoro identity block often produces exponential behavior consistent with the chaos bound:

$$\langle W(t)V(0)W(t)V(0)\rangle_\beta \sim 1 - \frac{\text{const}}{c} e^{\frac{2\pi}{\beta}t} + \cdots,$$

so the Lyapunov exponent is

$$\lambda_L = \frac{2\pi}{\beta}.$$

From the bulk side, this is described by gravitational shockwaves near the BTZ horizon (or, equivalently, by eikonal scattering). The key message for $AdS_3$ is that the relevant computation is often exactly the large-$c$ Virasoro vacuum block.

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9. Entanglement in $AdS_3/CFT_2$

Entanglement is where $AdS_3/CFT_2$ becomes both conceptually sharp and computationally concrete.

9.1 Ryu–Takayanagi in $AdS_3$

For a static state, the RT formula says

$$S_A = \frac{\text{Length}(\gamma_A)}{4G},$$

where $\gamma_A$ is the bulk geodesic anchored on the endpoints of interval $A$.

Using the geodesic length in global AdS gives

$$S_A = \frac{c}{3}\log\left(\frac{2}{\epsilon}\sin\frac{\Delta\phi}{2}\right).$$

9.2 Finite-temperature and rotating BTZ entanglement

For a thermal state (non-rotating), the CFT result is

$$S_A = \frac{c}{3}\log\left(\frac{\beta}{\pi\epsilon}\sinh\frac{\pi \ell_A}{\beta}\right),$$

where $\ell_A$ is the interval length on the line (or the corresponding angular size on the circle).

For the rotating case, left/right temperatures appear:

$$S_A = \frac{c}{6}\log\left(\frac{\beta_L}{\pi\epsilon}\sinh\frac{\pi \Delta x^+}{\beta_L}\right) + \frac{c}{6}\log\left(\frac{\beta_R}{\pi\epsilon}\sinh\frac{\pi \Delta x^-}{\beta_R}\right),$$

where $\Delta x^\pm$ are the separations in $x^\pm=t\pm \phi$.

The bulk statement is that the geodesic length in rotating BTZ splits naturally into left/right pieces.

9.3 Twist operators and why RT works so well in 2D

In a 2D CFT, the replica trick reduces entanglement entropy of one interval to a two-point function of twist operators. The twist operator has dimension

$$h_n=\bar h_n = \frac{c}{24}\left(n-\frac{1}{n}\right).$$

Taking $n\to 1$ and using large-$c$ factorization reproduces RT. This provides an explicit microscopic derivation (within the regime where the vacuum block dominates).

9.4 Multi-interval entanglement and phase transitions

For multiple intervals, RT predicts competing geodesic networks. In CFT language, this corresponds to different OPE channels of twist correlators, leading to sharp “phase transitions” in mutual information at large $c$.

This topic is a clean entry point to entanglement wedge reconstruction, quantum error correction, and the geometric meaning of factorization.

9.5 Kinematic space (optional): integral geometry of entanglement

In $AdS_3$, the space of boundary-anchored geodesics has a natural geometry (“kinematic space”), and entanglement entropy can be related to integral geometry (Crofton formulas). This becomes a powerful language for reconstructing bulk geometry from entanglement data and for understanding the emergence of bulk locality in 3D.

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10. $AdS_3$ gravity as Chern–Simons theory

This is one of the deepest structural simplifications of 3D gravity.

10.1 Gauge fields from triads and spin connection

In first-order form, define triads $e^a$ and spin connections $\omega^a$ ($a=0,1,2$). Then define two $sl(2,\mathbb{R})$ gauge fields:

$$A = \left(\omega^a + \frac{1}{\ell}e^a\right) L_a,\qquad \bar A = \left(\omega^a - \frac{1}{\ell}e^a\right) L_a.$$

A convenient $sl(2)$ basis is $L_{-1},L_0,L_{+1}$ with

$$[L_m,L_n]=(m-n)L_{m+n}.$$

The inverse map is

$$e = \frac{\ell}{2}(A-\bar A),\qquad \omega = \frac{1}{2}(A+\bar A).$$

The metric is recovered as a bilinear in the triad:

$$g_{\mu\nu} = \frac{1}{2}\mathrm{Tr}(e_\mu e_\nu),$$

with a trace convention fixed once and for all (different normalizations correspond to rescaling $k$ and the trace).

10.2 Chern–Simons action and level

The 3D Einstein action (with boundary terms) can be written as

$$S = S_{\text{CS}}[A]-S_{\text{CS}}[\bar A] + (\text{boundary terms}),$$

with

$$S_{\text{CS}}[A] = \frac{k}{4\pi}\int \mathrm{Tr}\left(A\wedge dA+\frac{2}{3}A\wedge A\wedge A\right),$$

and level

$$k=\frac{\ell}{4G}.$$

The Brown–Henneaux central charge becomes

$$c=6k,$$

consistent with $c=3\ell/(2G)$.

10.3 Boundary WZW theory, Drinfeld–Sokolov reduction, and Liouville theory

On manifolds with boundary, Chern–Simons theory induces a chiral WZW model on the boundary. Imposing Brown–Henneaux boundary conditions corresponds to a Drinfeld–Sokolov reduction of this WZW theory, which in turn yields Liouville theory as an effective boundary description.

This is one of the cleanest manifestations of how “gravity in the bulk” becomes “conformal dynamics on the boundary” in 3D.

10.4 Bañados data and Chern–Simons connections

In a gauge adapted to FG coordinates, one can write

$$A = b^{-1}(a\,dx^+ + \cdots)b + b^{-1}db,\qquad b=e^{\rho L_0},$$

with

$$a = \left(L_1 - \frac{2\pi}{k}\mathcal{L}(x^+)L_{-1}\right)dx^+,$$

and similarly

$$\bar a = \left(L_{-1} - \frac{2\pi}{k}\bar{\mathcal{L}}(x^-)L_{+1}\right)dx^-.$$

The functions $\mathcal{L}(x^+),\bar{\mathcal{L}}(x^-)$ are essentially $L(x^+),\bar L(x^-)$ up to normalization and encode the boundary stress tensor modes.

This is the most practical bridge between:

• (i) Bañados geometries,
• (ii) Virasoro charges,
• (iii) and holonomy characterizations of saddles.

10.5 BTZ as holonomies and smoothness conditions

In the Chern–Simons language, BTZ is characterized by the holonomy of $A$ and $\bar A$ around the Euclidean time and angular cycles.

Smoothness of the Euclidean geometry translates into a condition that the holonomy around the contractible cycle is trivial up to the center of the gauge group.

This turns thermodynamic relations (like the first law) into algebraic statements about flat connections.

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11. Higher-spin $AdS_3/CFT_2$ and $W$-algebras

Higher-spin holography in $AdS_3$ is unusually explicit because higher-spin gravity in 3D is also Chern–Simons theory.

11.1 $SL(N)\times SL(N)$ Chern–Simons and $W_N$ symmetry

Replace $sl(2)$ by $sl(N)$:

$$S = S_{\text{CS}}[A]-S_{\text{CS}}[\bar A],\qquad A,\bar A\in sl(N,\mathbb{R}).$$

With the principal embedding of $sl(2)$ into $sl(N)$, the asymptotic symmetry algebra becomes a $W_N$ algebra. In the large-$N$ limit one encounters $W_\infty$-type algebras and higher-spin algebras like $hs[\lambda]$.

11.2 Why $AdS_3$ higher-spin is a clean testing ground

In many proposals, the dual CFT is a well-controlled family (for example, large-$N$ limits of $W_N$ minimal models) with extended chiral symmetry. This allows:

• matching current spectra and operator dimensions,
• computing partition functions,
• and testing correlators beyond semiclassical gravity.

The field is also a laboratory for conceptual questions:

• What is “geometry” when the metric is gauge-dependent?
• How do black holes generalize when higher-spin gauge fields are present?

11.3 Higher-spin black holes and holonomy constraints

Higher-spin theories admit black-hole-like solutions characterized by higher-spin charges and chemical potentials. Thermodynamics and smoothness again become holonomy constraints, generalizing the BTZ story.

A key lesson is that in higher-spin gravity, “horizon area” is not gauge-invariant, so entropy must be defined more carefully (typically via holonomies or a generalized first law).

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12. Strings on $AdS_3$: NS–NS WZW, RR flux, D1–D5, and tensionless limits

Top-down holography in $AdS_3$ is dominated by strings on

$$AdS_3\times S^3\times M_4,\qquad M_4=T^4\ \text{or}\ K3.$$

The boundary dual is the D1–D5 CFT, a 2D $\mathcal{N}=(4,4)$ SCFT whose central charge is

$$c = 6N,\qquad N=Q_1Q_5$$

for $Q_1$ D1-branes and $Q_5$ D5-branes (in the simplest setup).

12.1 NS–NS flux and the $SL(2,\mathbb{R})_k$ WZW model

With pure NS–NS flux, the worldsheet theory on $AdS_3$ is the $SL(2,\mathbb{R})_k$ WZW model (together with an $SU(2)_k$ WZW model for $S^3$).

This is an exactly solvable CFT and leads to:

• explicit spectrum analysis (including discrete and continuous representations),
• “long strings” and continuum sectors,
• and analytic control over many observables.

A key structural feature is spectral flow, which generates physically distinct sectors and is essential for modular invariance and for describing BTZ states.

12.2 RR flux and integrability

With pure RR flux, the worldsheet is not a simple WZW model. Instead, one often uses:

• Green–Schwarz / hybrid formalisms,
• integrability to constrain the exact spectrum and S-matrix (see Section 13).

Mixed flux interpolates between these regimes and retains integrable structure in many cases.

12.3 The D1–D5 CFT and the symmetric orbifold point

At a special point in moduli space, the D1–D5 CFT is the symmetric product orbifold

$$\mathrm{Sym}^N(M_4) = \frac{(M_4)^N}{S_N}.$$

This point is solvable and supports explicit computations of:

• BPS spectra,
• elliptic genera and supersymmetric indices,
• many correlators using covering surface technology.

Moving away from the orbifold point corresponds to turning on exactly marginal deformations; in the bulk, these correspond to changing stringy moduli and (for mixed flux) interpolating between NS–NS and RR regimes.

12.4 Tensionless limits and “more exact” $AdS_3/CFT_2$

A major modern theme is that some $AdS_3$ string backgrounds admit tensionless or near-tensionless regimes where:

• stringy symmetry becomes enormous,
• correlators simplify dramatically,
• and one can match bulk and boundary data in unusually explicit detail.

This strongly influences current thinking about what an “exactly solvable” holographic pair might look like.

12.5 Modern amplitude program in $AdS_3\times S^3$

A recent line of work extracts CFT data (and sometimes bulk string amplitudes) from protected or controlled observables in $AdS_3\times S^3$ backgrounds, often emphasizing:

• exact worldsheet control in NS–NS backgrounds,
• bootstrap constraints,
• and “Virasoro–Shapiro” type amplitudes that play a role analogous to the flat-space Virasoro–Shapiro amplitude.

This is a fast-moving topic; see the reading list for recent entry points.

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13. Integrability in $AdS_3/CFT_2$

Integrability has become central for extracting nontrivial dynamical data at strong coupling, especially for RR and mixed-flux regimes.

13.1 What integrability gives you

In integrable AdS/CFT systems, one can often determine:

• exact dispersion relations,
• factorized worldsheet scattering,
• exact S-matrices constrained by symmetry,
• Bethe ansatz equations for finite-size spectra,
• thermodynamic Bethe ansatz (TBA) and Y-systems,
• and in some cases a quantum spectral curve (QSC) formulation.

13.2 What is special (and challenging) in $AdS_3$

Compared with $AdS_5/CFT_4$, $AdS_3$ integrability has extra subtleties:

• Massless modes appear and require special care in scattering, crossing, and finite-size effects.
• There are multiple background families ($T^4$ vs $S^3\times S^3\times S^1$, mixed flux).
• The relation between integrability variables and the D1–D5 CFT operator spectrum can be more intricate.

Nevertheless, the subject has reached a mature stage, with recent reviews and systematic frameworks.

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14. Solvable irrelevant deformations: $T\bar T$ and friends

Two-dimensional QFT admits special irrelevant deformations that remain solvable in surprising ways.

14.1 The $T\bar T$ operator

Define the (Euclidean) stress tensor components $T(z)$ and $\bar T(\bar z)$ and the trace $\Theta(z,\bar z)$. The $T\bar T$ composite operator is

$$T\bar T \equiv T(z)\bar T(\bar z) - \Theta(z,\bar z)^2.$$

Deforming a CFT by

$$S_\lambda = S_{\text{CFT}} + \lambda \int d^2x\, (T\bar T)(x)$$

produces a theory with remarkable solvability properties, including an exactly solvable flow of finite-volume energy levels.

14.2 Energy spectrum flow (key PDE and closed-form solution)

On a circle of circumference $L$, for an eigenstate with momentum $P_n$, the deformed energy $E_n(\lambda,L)$ obeys a Burgers-type equation:

$$\partial_\lambda E_n(\lambda,L) = -\frac{1}{2}\left(E_n(\lambda,L)\,\partial_L E_n(\lambda,L) + \frac{P_n^2}{L}\right).$$

Solving this PDE with initial condition $E_n(0,L)=E_n^{(0)}(L)$ gives a closed-form expression (one common convention):

$$E_n(\lambda,L) = \frac{L}{2\lambda} \left( 1-\sqrt{1-\frac{2\lambda}{L}E_n^{(0)}(L)+\left(\frac{\lambda P_n}{L}\right)^2} \right).$$

The square-root structure is responsible for the characteristic UV behavior of the theory and its “Hagedorn-like” density of states at high energies (for one sign of $\lambda$).

14.3 Cutoff $AdS_3$ interpretation (holographic viewpoint)

A widely used holographic interpretation is:

• a $T\bar T$-deformed CFT is dual to gravity in $AdS_3$ with a finite radial cutoff and modified boundary conditions.

Even when the precise dual statement depends on conventions and matter content, $T\bar T$ is now standard for exploring:

• quasi-local holography,
• finite-cutoff observables,
• and UV nonlocality in a controlled setting.

14.4 Other solvable deformations

Other deformations include $J\bar T$ and related current-stress couplings, which often preserve part of the chiral algebra and lead to rich solvable structures. They appear naturally in warped $AdS_3$/CFT$_2$ and non-Lorentz-invariant holography.

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15. “Pure” $AdS_3$ gravity, modular bootstrap, and the sum over saddles

A famously subtle question:

Does “pure” 3D Einstein gravity with negative cosmological constant define a consistent quantum theory whose only degrees of freedom are boundary gravitons and BTZ black holes?

The short answer is: semiclassically, yes; nonperturbatively, this remains subtle and is an active research frontier.

15.1 What is solidly established (the “universal semiclassical layer”)

There is a robust set of statements that hold for any holographic CFT with large $c$ and a sparse light spectrum. These are the parts of $AdS_3/CFT_2$ that are essentially “as established” as higher-dimensional AdS/CFT:

• Brown–Henneaux: asymptotic symmetry is $\mathrm{Vir}\times\overline{\mathrm{Vir}}$ with $c=3\ell/(2G)$.
• BTZ thermodynamics matches Cardy.
• RT/HRT reproduces universal entanglement patterns at leading order.
• Heavy-light correlators exhibit emergent thermality controlled by the vacuum Virasoro block.

In this regime, “$AdS_3$ gravity” means an effective theory describing these universal features; the dual “$CFT_2$” means any large-$c$ CFT with the right sparseness and factorization properties.

15.2 Why “pure gravity” is much sharper in 3D than in higher dimensions

In higher dimensions, “pure gravity” is clearly not UV complete, but string theory provides the UV completion, and we mostly ask effective questions.

In 3D, the situation is sharper because:

• there are no local gravitons, so one might hope for an exact quantum theory with a relatively small Hilbert space,
• the classical phase space is essentially “boundary data,” suggesting quantization might be tractable,
• but the Euclidean path integral includes many saddles (different topologies), and it is nontrivial to decide what should be included and with what measure.

This leads to questions that are almost unique to $AdS_3$:

• Does the bulk path integral factorize as a single CFT partition function?
• Or does it compute an average over an ensemble of CFTs (as in JT gravity)?
• Can one define a consistent Hilbert space and operator algebra for “pure” gravity?

15.3 The Maloney–Witten sum and the “extremal CFT” tension

A canonical attempt to define pure gravity is:

• sum over handlebody saddles compatible with a given boundary torus (and more generally higher genus surfaces),
• include one-loop determinants.

On the torus, this produces a modular-invariant object reminiscent of a Poincaré series built from the vacuum character. However, the resulting spectral density is not obviously that of a single unitary CFT (issues include continuous spectra/negative degeneracies in certain regimes).

This motivates two lines of inquiry:

• restrict or modify the sum over saddles,
• or interpret the answer as something like an ensemble average.

15.4 Modern viewpoint: ensemble holography and TQFT gravity

Motivated by Euclidean wormholes and by successes in JT gravity, much recent work explores the idea that:

• semiclassical $AdS_3$ gravity computations are reproduced by averages over CFT data (density of states and OPE statistics),
• and that the bulk can be reformulated in terms of a topological theory (“Virasoro TQFT” and related structures) that computes amplitudes algorithmically on fixed topologies.

At the same time, there are important constraints:

• certain “sub-threshold” observables may not show ensemble averaging,
• and the relationship between fixed-topology amplitudes, sums over topologies, and CFT factorization is highly nontrivial.

15.5 What is not settled (open problems worth working on)

Here is a non-exhaustive list of concrete unsettled topics that are genuinely research-active:

1. Defining “pure $AdS_3$ gravity” nonperturbatively
   What is the correct set of bulk saddles/topologies? How are they weighted? What is the correct boundary condition prescription?

2. Factorization vs wormholes vs ensembles
   When do connected Euclidean wormholes contribute? Which observables show apparent ensemble averaging, and can this be reconciled with a single-unitary-CFT dual?

3. Higher-genus partition functions and the mapping class group
   Can one compute higher-genus gravity partition functions exactly in $c$ (not just semiclassically)? How do these objects behave under mapping class group actions?

4. Microscopic interpretation of the boundary graviton sector
   When does the vacuum Virasoro module accurately capture low-lying bulk states? How are null states, additional conserved currents, or extended symmetry reflected in bulk physics?

5. Exact dynamics in stringy $AdS_3$
   For D1–D5, can one systematically determine dynamical CFT data away from the orbifold point and match to bulk string scattering?

6. Interplay between $T\bar T$/cutoff holography and the 3D gravity path integral
   Can finite-cutoff boundary conditions regularize or reorganize the sum over topologies? What is the precise relation to solvable deformations in the boundary theory?

7. Quantum chaos and fine-grained spectral statistics of BTZ microstates
   How universal is random-matrix behavior? How does it emerge from Virasoro symmetry and modularity, and how does it depend on which observables you probe?

If you want problems that are “doable but deep,” start with (2), (3), or (7) and try to reproduce key results in the papers listed in Section 17.

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16. Exercises and mini-projects

These are designed so that (i) each exercise teaches a standard computational move in $AdS_3/CFT_2$, and (ii) the collection forms a coherent “training loop.”

Exercise 1: Derive $c=\frac{3\ell}{2G}$ from the trace anomaly

Use the holographic stress tensor in FG gauge to show that

$$\langle T^i_{\ i}\rangle = \frac{c}{24\pi}R[g^{(0)}]$$

with $c=\frac{3\ell}{2G}$.

Solution

In FG gauge,

$$ds^2=\ell^2\frac{dz^2}{z^2}+\frac{1}{z^2}g_{ij}(x,z)\,dx^i dx^j, \qquad g_{ij}=g^{(0)}_{ij}+z^2 g^{(2)}_{ij}+O(z^4).$$

The renormalized holographic stress tensor in $d=2$ is

$$\langle T_{ij}\rangle = \frac{\ell}{8\pi G}\left(g^{(2)}_{ij}-g^{(0)}_{ij}\,\mathrm{Tr}\,g^{(2)}\right), \qquad \mathrm{Tr}\,g^{(2)}\equiv g^{(0)ij}g^{(2)}_{ij}.$$

Taking the trace,

$$\langle T^i_{\ i}\rangle = \frac{\ell}{8\pi G}\left(\mathrm{Tr}\,g^{(2)}-2\,\mathrm{Tr}\,g^{(2)}\right) = -\frac{\ell}{8\pi G}\,\mathrm{Tr}\,g^{(2)}.$$

The FG constraints from the Einstein equations imply

$$\mathrm{Tr}\,g^{(2)} = -\frac{1}{2}R[g^{(0)}].$$

Therefore

$$\langle T^i_{\ i}\rangle = \frac{\ell}{16\pi G}R[g^{(0)}].$$

Comparing with the 2D CFT anomaly $\langle T^i_{\ i}\rangle = \frac{c}{24\pi}R$, we get

$$\frac{c}{24\pi} = \frac{\ell}{16\pi G} \quad\Rightarrow\quad c=\frac{3\ell}{2G}.$$

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Exercise 2: Show that Cardy reproduces the rotating BTZ entropy

Starting with

$$L_0-\frac{c}{24}=\frac{\ell M+J}{2},\qquad \bar L_0-\frac{c}{24}=\frac{\ell M-J}{2},$$

and the BTZ relations

$$M=\frac{r_+^2+r_-^2}{8G\ell^2},\qquad J=\frac{r_+r_-}{4G\ell},$$

show that Cardy gives $S_{\text{BH}}=2\pi r_+/(4G)$.

Solution

Compute

$$\ell M+J=\ell\frac{r_+^2+r_-^2}{8G\ell^2}+\frac{r_+r_-}{4G\ell} = \frac{r_+^2+r_-^2+2r_+r_-}{8G\ell} = \frac{(r_+ + r_-)^2}{8G\ell}.$$

Similarly

$$\ell M-J = \frac{r_+^2+r_-^2-2r_+r_-}{8G\ell} = \frac{(r_+ - r_-)^2}{8G\ell}.$$

Therefore

$$L_0-\frac{c}{24}=\frac{(r_+ + r_-)^2}{16G\ell},\qquad \bar L_0-\frac{c}{24}=\frac{(r_+ - r_-)^2}{16G\ell}.$$

Cardy gives

$$S_{\text{Cardy}} = 2\pi\sqrt{\frac{c}{6}\left(L_0-\frac{c}{24}\right)} + 2\pi\sqrt{\frac{c}{6}\left(\bar L_0-\frac{c}{24}\right)}.$$

Using $c=\frac{3\ell}{2G}$,

$$\sqrt{\frac{c}{6}\left(L_0-\frac{c}{24}\right)} = \sqrt{\frac{1}{6}\frac{3\ell}{2G}\cdot \frac{(r_+ + r_-)^2}{16G\ell}} = \frac{r_+ + r_-}{8G}.$$

Similarly the right-moving piece gives $(r_+ - r_-)/(8G)$.

Hence

$$S_{\text{Cardy}} = 2\pi\left(\frac{r_+ + r_-}{8G}+\frac{r_+ - r_-}{8G}\right) = \frac{2\pi r_+}{4G} = S_{\text{BH}}.$$

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Exercise 3: Derive the thermal two-point function on the cylinder

Using the map $z=e^{\frac{2\pi}{\beta}w}$ from the cylinder ($w=\sigma+i\tau$, $\tau\sim\tau+\beta$) to the plane, derive

$$\langle \mathcal{O}(w,\bar w)\mathcal{O}(0,0)\rangle_\beta = \left(\frac{\pi/\beta}{\sin\left(\frac{\pi}{\beta}w\right)}\right)^{2h} \left(\frac{\pi/\beta}{\sin\left(\frac{\pi}{\beta}\bar w\right)}\right)^{2\bar h}.$$

Solution

On the plane, the two-point function is fixed by conformal invariance:

$$\langle \mathcal{O}(z,\bar z)\mathcal{O}(0,0)\rangle = \frac{1}{z^{2h}\bar z^{2\bar h}}.$$

Under a conformal map $z=z(w)$, a primary transforms as

$$\mathcal{O}(w,\bar w) = \left(\frac{dz}{dw}\right)^h\left(\frac{d\bar z}{d\bar w}\right)^{\bar h}\mathcal{O}(z,\bar z).$$

Take $z = e^{\frac{2\pi}{\beta}w}$, so

$$\frac{dz}{dw} = \frac{2\pi}{\beta}e^{\frac{2\pi}{\beta}w} = \frac{2\pi}{\beta}z.$$

Then

$$\begin{aligned} \langle \mathcal{O}(w,\bar w)\mathcal{O}(0,0)\rangle_\beta &= \left(\frac{dz}{dw}\right)^h\left(\frac{d\bar z}{d\bar w}\right)^{\bar h} \langle \mathcal{O}(z,\bar z)\mathcal{O}(1,1)\rangle \\ &= \left(\frac{2\pi}{\beta}\right)^{h+\bar h} \frac{z^h\bar z^{\bar h}}{(z-1)^{2h}(\bar z-1)^{2\bar h}}. \end{aligned}$$

Now use

$$z-1 = e^{\frac{\pi}{\beta}w}\left(e^{\frac{\pi}{\beta}w}-e^{-\frac{\pi}{\beta}w}\right) = 2e^{\frac{\pi}{\beta}w}\sinh\left(\frac{\pi}{\beta}w\right),$$

so

$$\frac{z^h}{(z-1)^{2h}} = \frac{e^{\frac{2\pi h}{\beta}w}}{\left(2e^{\frac{\pi}{\beta}w}\sinh\left(\frac{\pi}{\beta}w\right)\right)^{2h}} = \frac{1}{(2\sinh\left(\frac{\pi}{\beta}w\right))^{2h}}.$$

In Euclidean signature one often writes the result with $\sin$ rather than $\sinh$ depending on conventions for $w=\sigma+i\tau$; the standard compact expression is

$$\left(\frac{\pi/\beta}{\sin\left(\frac{\pi}{\beta}w\right)}\right)^{2h},$$

and similarly for the antiholomorphic part. Analytically continuing $\tau\to it$ then produces the $\sinh$ form in Lorentzian time.

---

Exercise 4: One-loop vacuum character from boundary gravitons

Explain why the one-loop partition function around thermal $AdS_3$ is

$$Z_{\text{1-loop}}(\tau,\bar\tau) \sim |q|^{-c/12}\prod_{n=2}^\infty \frac{1}{|1-q^n|^2}.$$

Solution

Brown–Henneaux boundary conditions imply that the physical excitations around global $AdS_3$ are boundary gravitons generated by Virasoro descendants of the vacuum.

In the CFT vacuum module, the holomorphic descendants are created by $L_{-n}$ with $n\ge 2$; the $n=1$ mode is a global $SL(2,\mathbb{R})$ generator and does not create an independent state.

A state with occupation numbers $\{N_n\}$ has

$$L_0 = -\frac{c}{24} + \sum_{n=2}^\infty n\,N_n,$$

so tracing over all $N_n\in\mathbb{Z}_{\ge 0}$ gives the vacuum character

$$\chi_{\text{vac}}(q) = q^{-\frac{c}{24}}\prod_{n=2}^\infty \frac{1}{1-q^n}.$$

Including both left and right sectors yields

$$Z_{\text{1-loop}} \propto \chi_{\text{vac}}(q)\,\overline{\chi_{\text{vac}}(q)} = |q|^{-\frac{c}{12}}\prod_{n=2}^\infty \frac{1}{|1-q^n|^2},$$

up to an overall normalization coming from zero-modes and the classical saddle action.

---

Exercise 5: Heavy–light vacuum block and effective temperature

In the heavy–light limit, show that the holomorphic vacuum Virasoro block behaves as

$$\mathcal{F}_{\text{vac}}(z)\approx \left(\frac{\alpha z^{\frac{\alpha-1}{2}}}{1-z^\alpha}\right)^{2h_L}, \qquad \alpha=\sqrt{1-\frac{24h_H}{c}},$$

and interpret the $\alpha=i\lambda$ case as thermality with $\beta_{\text{eff}}=2\pi/\lambda$.

Solution

In the semiclassical regime $c\to\infty$ with $h_H\sim O(c)$ and $h_L\ll c$, the heavy operators source a classical expectation value of the stress tensor:

$$\langle T(z)\rangle_H \sim \frac{h_H}{z^2}$$

(in a coordinate system where the heavy insertions are at $0$ and $\infty$).

The key move is that a holomorphic coordinate transformation $z\to w(z)$ can “uniformize” this background so that $\langle T(w)\rangle=0$. Using the transformation law

$$T(w) = \left(\frac{dz}{dw}\right)^2 T(z) + \frac{c}{12}\{z,w\},$$

one finds a solution of the form

$$w = z^\alpha,\qquad \alpha=\sqrt{1-\frac{24h_H}{c}}.$$

In this coordinate, the vacuum block reduces to a global block in the $w$-plane. For a pair of light operators, this yields

$$\mathcal{F}_{\text{vac}}(z) \propto \left(w'(1)\right)^{h_L}\left(w'(z)\right)^{h_L}\frac{1}{(w(1)-w(z))^{2h_L}}.$$

Compute

$$w(z)=z^\alpha,\qquad w'(z)=\alpha z^{\alpha-1}.$$

Then

$$w(1)-w(z)=1-z^\alpha,$$

so

$$\mathcal{F}_{\text{vac}}(z) \propto \left(\alpha\cdot 1^{\alpha-1}\right)^{h_L}\left(\alpha z^{\alpha-1}\right)^{h_L} \frac{1}{(1-z^\alpha)^{2h_L}} = \left(\frac{\alpha z^{\frac{\alpha-1}{2}}}{1-z^\alpha}\right)^{2h_L}.$$

If $h_H>c/24$, then $\alpha=i\lambda$ with $\lambda=\sqrt{\frac{24h_H}{c}-1}$, and

$$z^\alpha = e^{i\lambda\log z}.$$

This introduces an effective periodicity in Euclidean time, corresponding to an effective inverse temperature

$$\beta_{\mathrm{eff}}=\frac{2\pi}{\lambda},$$

matching the BTZ relation between $h_H$ and left-moving temperature.

---

Exercise 6: Holonomy condition for BTZ in Chern–Simons language

Explain how smoothness of Euclidean BTZ becomes a statement that the holonomy around the contractible cycle is trivial (up to the center) for the $SL(2)$ connections $A,\bar A$.

Solution

In Euclidean signature, BTZ is a solid torus. One cycle of the boundary torus is contractible in the bulk; demanding smoothness means that going around that cycle should not create a conical singularity.

In Chern–Simons language, the geometry is encoded by flat connections $A,\bar A$. For a flat connection, the only gauge-invariant data around a nontrivial cycle $\gamma$ is the holonomy

$$\mathrm{Hol}_\gamma(A) = \mathcal{P}\exp\left(\oint_\gamma A\right).$$

If $\gamma$ is contractible in the bulk, smoothness requires that the holonomy be trivial up to an element of the center of the gauge group (because a central element acts trivially on adjoint fields and does not produce a physical singularity):

$$\mathrm{Hol}_{\text{contractible}}(A)\in \text{center},\qquad \mathrm{Hol}_{\text{contractible}}(\bar A)\in \text{center}.$$

For $SL(2)$, this means the eigenvalues of the holonomy matrix are fixed (typically to $\pm 1$), which imposes algebraic relations between:

• the charges (mass and angular momentum),
• and the chemical potentials (temperature and angular velocity).

These relations reproduce the usual BTZ thermodynamics. In higher-spin generalizations, the same logic applies but with more holonomy constraints, encoding the generalized first law.

---

Exercise 7 (bonus): Recover the Bañados metric from Chern–Simons boundary conditions

Start from the Drinfeld–Sokolov form

$$a = \left(L_1 - \frac{2\pi}{k}\mathcal{L}(x^+)L_{-1}\right)dx^+,\qquad \bar a = \left(L_{-1} - \frac{2\pi}{k}\bar{\mathcal{L}}(x^-)L_{+1}\right)dx^-,$$

with $A=b^{-1}ab+b^{-1}db$, $\bar A=b\bar a b^{-1}+bdb^{-1}$ and $b=e^{\rho L_0}$. Show that the resulting metric takes the Bañados form with $L\propto\mathcal{L}$.

Solution

The essential steps are conceptual rather than computationally heavy:

1. Use

$$A = b^{-1}ab+b^{-1}db,\qquad \bar A = b\bar a b^{-1}+bdb^{-1},\qquad b=e^{\rho L_0}.$$

This gauge choice builds in the $e^{\pm\rho}$ behavior that matches FG falloffs.

2. Compute the triad

$$e = \frac{\ell}{2}(A-\bar A).$$

3. Form the metric

$$g_{\mu\nu} = \frac{1}{2}\mathrm{Tr}(e_\mu e_\nu).$$

Because $a$ has only a $dx^+$ component and $\bar a$ only a $dx^-$ component, the resulting metric has the structure

• $g_{++}$ controlled by $\mathcal{L}(x^+)$,
• $g_{--}$ controlled by $\bar{\mathcal{L}}(x^-)$,
• and $g_{+-}$ controlled by the $e^{2\rho}$ leading term and an $e^{-2\rho}$ tail proportional to $\mathcal{L}\bar{\mathcal{L}}$.

With an appropriate trace convention, the resulting metric matches

$$ds^2 = \ell^2 d\rho^2 + \left(e^{2\rho} + \frac{1}{4}e^{-2\rho}L(x^+)\bar L(x^-)\right)dx^+dx^- + L(x^+)dx^{+2} + \bar L(x^-)dx^{-2},$$

with

$$L(x^+) \propto \frac{2\pi}{k}\mathcal{L}(x^+),\qquad \bar L(x^-) \propto \frac{2\pi}{k}\bar{\mathcal{L}}(x^-),$$

and the proportionality factor fixed by your choice of trace normalization.

The key physical point is that Drinfeld–Sokolov boundary conditions are precisely the Chern–Simons implementation of Brown–Henneaux boundary conditions, so the most general solution must reproduce the Bañados family.

---

Exercise 8 (bonus): $T\bar T$ energy formula from the Burgers equation

Solve the Burgers-type flow equation for $E_n(\lambda,L)$ and show it yields

$$E_n(\lambda,L) = \frac{L}{2\lambda} \left( 1-\sqrt{1-\frac{2\lambda}{L}E_n^{(0)}(L)+\left(\frac{\lambda P_n}{L}\right)^2} \right).$$

Solution

A standard route is the method of characteristics.

Write the PDE as

$$\partial_\lambda E + \frac{1}{2}E\,\partial_L E = -\frac{P^2}{2L}.$$

Treat $E(\lambda,L)$ along characteristic curves $L(\lambda)$ defined by

$$\frac{dL}{d\lambda} = \frac{1}{2}E(\lambda,L(\lambda)).$$

Then along the characteristic,

$$\frac{dE}{d\lambda} = \partial_\lambda E + \frac{dL}{d\lambda}\partial_L E = -\frac{P^2}{2L}.$$

Now consider the combination

$$\frac{d}{d\lambda}(E^2 - P^2) = 2E\frac{dE}{d\lambda} = -E\frac{P^2}{L}.$$

Using $dL/d\lambda = E/2$ gives

$$\frac{d}{d\lambda}(E^2 - P^2) = -2\frac{P^2}{L}\frac{dL}{d\lambda}.$$

This implies a first integral of motion along characteristics of the form

$$E^2 - P^2 + 2\frac{P^2}{L}L = \text{const},$$

and more carefully one finds that the quantity

$$\left(1-\frac{2\lambda}{L}E\right)^2 - \left(\frac{\lambda P}{L}\right)^2$$

is conserved along the flow when matched to the initial condition at $\lambda=0$.

Imposing $E(0,L)=E^{(0)}(L)$ yields the square-root solution

$$E(\lambda,L) = \frac{L}{2\lambda} \left( 1-\sqrt{1-\frac{2\lambda}{L}E^{(0)}(L)+\left(\frac{\lambda P}{L}\right)^2} \right),$$

with the branch chosen so that $E(\lambda,L)\to E^{(0)}(L)$ as $\lambda\to 0$.

(Exact sign conventions vary in the literature; the essential physics is the universal square-root structure.)

---

17. Annotated reading list

This is not a complete bibliography. It is a curated set of entry points that are:

• historically important,
• technically useful,
• and representative of modern research directions.

The format is designed to work well inside Astro/Starlight pages (HTML inside Markdown).

17.1 CFT and 2D conformal symmetry

[1] P. Di Francesco, P. Mathieu and D. Sénéchal, Conformal Field Theory (Springer, New York, 1997). — The standard CFT2 “bible.” Use this as the default reference for definitions and conventions.

[2] P. Ginsparg, Applied Conformal Field Theory, arXiv:hep-th/9108028. — A classic lecture-note style introduction; excellent for building intuition.

[3] J. Polchinski, String Theory, Vol. 1 (Cambridge University Press, 1998). — For CFT as worldsheet physics; very useful when moving to stringy $AdS_3$.

[4] S. Rychkov, EPFL Lectures on Conformal Field Theory in D≥3 Dimensions (2016). — Not 2D-specific, but a clean conceptual entry to bootstrap logic.

[5] A. A. Belavin, A. M. Polyakov and A. B. Zamolodchikov, Infinite Conformal Symmetry in Two-Dimensional Quantum Field Theory, Nucl. Phys. B 241 (1984) 333. — The original BPZ paper.

[6] J. Cardy, Operator Content of Two-Dimensional Conformally Invariant Theories, Nucl. Phys. B 270 (1986) 186. — Modular invariance and state counting in early form.

[7] A. B. Zamolodchikov, Conformal symmetry in two-dimensional space: Recursion representation of conformal block, Theor. Math. Phys. 73 (1987) 1088. — The recursion relation behind practical Virasoro block computations.

[8] J. Teschner, Liouville theory revisited, JHEP 05 (2001) 042 [arXiv:hep-th/0104158]. — A standard entry point to quantum Liouville theory and its exact bootstrap.

[9] P. Kraus and F. Larsen, Holographic gravitational anomalies, JHEP 01 (2006) 022 [arXiv:hep-th/0508218]. — Useful for anomalies and chiral CFT aspects in $AdS_3$ setups.

[10] S. Hellerman, A Universal Inequality for CFT and Quantum Gravity, JHEP 08 (2011) 130 [arXiv:0902.2790]. — A landmark modular bootstrap bound, extremely relevant for “pure gravity” discussions.

17.2 $AdS_3$ gravity, Brown–Henneaux, and holographic renormalization

[11] M. Bañados, C. Teitelboim and J. Zanelli, The Black Hole in Three Dimensional Space Time, Phys. Rev. Lett. 69 (1992) 1849 [arXiv:hep-th/9204099]. — The original BTZ paper.

[12] J. D. Brown and M. Henneaux, Central Charges in the Canonical Realization of Asymptotic Symmetries: An Example from Three-Dimensional Gravity, Commun. Math. Phys. 104 (1986) 207. — The origin of Virasoro symmetry and $c=\frac{3\ell}{2G}$.

[13] M. Bañados, Three-dimensional quantum geometry and black holes, arXiv:hep-th/9901148. — A clear review of the Bañados family, CFT–geometry map, and quantization ideas.

[14] K. Skenderis, Lecture notes on holographic renormalization, JHEP 08 (2002) 016 [arXiv:hep-th/0209067]. — The standard reference for holographic stress tensors and anomalies.

[15] M. Henneaux and C. Teitelboim, Quantum Mechanics of Fundamental Systems 2 (Plenum, 1987). — Early canonical approaches; useful historical context for 3D gravity.

[16] S. Carlip, The (2+1)-Dimensional Black Hole, Class. Quant. Grav. 12 (1995) 2853 [arXiv:gr-qc/9506079]. — A broad review including entropy and quantization ideas.

[17] O. Coussaert, M. Henneaux and P. van Driel, The Asymptotic Dynamics of Three-Dimensional Einstein Gravity with a Negative Cosmological Constant, Class. Quant. Grav. 12 (1995) 2961 [arXiv:gr-qc/9506019]. — Classic connection between $AdS_3$ gravity, WZW, and Liouville dynamics.

[18] E. Witten, Three-Dimensional Gravity Revisited, arXiv:0706.3359. — Conceptual discussion of what “pure” $AdS_3$ gravity might mean.

17.3 Virasoro blocks, semiclassical physics, and chaos

[19] A. L. Fitzpatrick, J. Kaplan and M. T. Walters, Virasoro Conformal Blocks and Thermality from Classical Background Fields, JHEP 11 (2015) 200 [arXiv:1501.05315]. — A clean derivation of heavy–light thermality via the Schwarzian transformation.

[20] D. A. Roberts and D. Stanford, Diagnosing Chaos Using Four-Point Functions in Two-Dimensional Conformal Field Theory, Phys. Rev. Lett. 115 (2015) 131603 [arXiv:1412.5123]. — Classic paper connecting large-$c$ Virasoro blocks to chaos in CFT2/$AdS_3$.

[21] E. Perlmutter, Virasoro conformal blocks in closed form, JHEP 08 (2015) 088 [arXiv:1502.07742]. — Practical expansions/representations of Virasoro blocks; useful for computations.

17.4 Pure gravity, modular bootstrap, and ensemble/TQFT perspectives

[22] A. Maloney and E. Witten, Quantum Gravity Partition Functions in Three Dimensions, JHEP 02 (2010) 029 [arXiv:0712.0155]. — The classic “sum over saddles on the torus” attempt; essential background for pure gravity discussions.

[23] T. Hartman, C. A. Keller and B. Stoica, Universal Spectrum of 2d Conformal Field Theory in the Large c Limit, JHEP 09 (2014) 118 [arXiv:1405.5137]. — How modular invariance + sparseness implies universal holographic thermodynamics.

[24] B. Benjamin, S. Collier and A. Maloney, Pure Gravity and its Ensemble of CFTs, JHEP 10 (2020) 112 [arXiv:2006.02494]. — A modern framing of pure gravity in terms of an ensemble interpretation.

[25] J. Cotler and K. Jensen, AdS$_3$ gravity and random CFT, JHEP 04 (2021) 033 [arXiv:2006.08648]. — Wormholes, random matrix behavior, and ensemble interpretations in $AdS_3$.

[26] J. Chandra, S. Collier, T. Hartman and A. Maloney, Semiclassical 3D gravity as an average of large-c CFTs, JHEP 12 (2022) 069 [arXiv:2203.06511]. — Relates semiclassical gravity computations to averages over CFT data.

[27] J.-M. Schlenker and E. Witten, No Ensemble Averaging Below the Black Hole Threshold, JHEP 07 (2022) 143 [arXiv:2202.01372]. — Important constraints on where ensemble behavior can appear.

[28] S. Collier, L. Eberhardt and M. Zhang, Solving 3d Gravity with Virasoro TQFT, SciPost Phys. 15 (2023) 151 [arXiv:2304.13650]. — Reformulates 3D gravity in terms of Virasoro TQFT and provides algorithmic computations on fixed topology.

[29] S. Collier, L. Eberhardt and M. Zhang, 3d gravity from Virasoro TQFT: Holography, wormholes and knots, SciPost Phys. 17 (2024) 134 [arXiv:2401.13900]. — Follow-up exploring multi-boundary wormholes and hyperbolic 3-manifolds.

[30] A. Dymarsky and A. Shapere, TQFT gravity and ensemble holography, arXiv:2405.20366. — A general derivation/motivation for TQFT-gravity ↔ ensemble-CFT duality.

[31] T. Hartman, Triangulating quantum gravity in AdS$_3$, arXiv:2507.12696. — A modern triangulation/topological approach to exact $AdS_3$ gravity amplitudes.

[32] A. Belin, A. Maloney and F. Seefeld, A measure on the space of CFTs and pure 3D gravity, arXiv:2509.04554. — A “maximum ignorance” measure on CFT space; tests pure-gravity-as-ensemble ideas.

[33] A. Belin, A universal sum over topologies in 3d gravity, arXiv:2601.07906. — Recent work on organizing the sum over topologies and related statistical bootstrap ideas.

[34] A. Barbar, Automorphism-weighted ensembles from TQFT gravity, arXiv:2511.04311. — Develops measures/weights in TQFT-gravity ensemble constructions.

17.5 Higher-spin $AdS_3/CFT_2$

[35] M. R. Gaberdiel and R. Gopakumar, An AdS$_3$ Dual for Minimal Model CFTs, Phys. Rev. D 83 (2011) 066007 [arXiv:1011.2986]. — The classic proposal for higher-spin/$W_N$ minimal model duality.

[36] M. Ammon, M. Gutperle, P. Kraus and E. Perlmutter, Black holes in three-dimensional higher spin gravity: a review, Class. Quant. Grav. 30 (2013) 214001 [arXiv:1208.5182]. — A solid entry point to higher-spin black holes and holonomy thermodynamics.

[37] A. Campoleoni, S. Fredenhagen, S. Pfenninger and S. Theisen, Asymptotic symmetries of three-dimensional gravity coupled to higher-spin fields, JHEP 11 (2010) 007 [arXiv:1008.4744]. — Derives $W$-algebras from higher-spin $AdS_3$ boundary conditions.

17.6 Stringy $AdS_3$, D1–D5, worldsheet methods, and modern amplitudes

[38] D. Kutasov and N. Seiberg, More Comments on String Theory on AdS$_3$, JHEP 04 (1999) 008 [arXiv:hep-th/9903219]. — Early and influential: long strings, worldsheet subtleties, and $AdS_3$ string physics.

[39] J. Maldacena and H. Ooguri, Strings in AdS$_3$ and the SL(2,R) WZW Model, JHEP 07 (2001) 049 [arXiv:hep-th/0001053]. — A foundational reference for worldsheet $AdS_3$ in the NS–NS regime.

[40] J. Maldacena and H. Ooguri, Strings in AdS$_3$ and the SL(2,R) WZW Model. Part 3: Correlation Functions, Phys. Rev. D 65 (2002) 106006 [arXiv:hep-th/0111180]. — Technical but essential for worldsheet correlators and spectral flow sectors.

[41] D. David, A. Mandal and S. Wadia, D1/D5 System and AdS$_3$/CFT$_2$, Phys. Lett. B 552 (2003) 273 [arXiv:hep-th/0203048]. — A standard review of the D1–D5 system and its AdS$_3$/CFT$_2$ interpretation.

[42] A. Dabholkar, Exact counting of black hole microstates, arXiv:hep-th/0409148. — Reviews microstate counting technology relevant for D1–D5 and related systems.

[43] M. R. Gaberdiel and R. Gopakumar, Tensionless string spectra on AdS$_3$, JHEP 05 (2015) 085 [arXiv:1501.07274]. — String spectra near tensionless points; good for the “large symmetry” viewpoint.

[44] L. Eberhardt, Symmetries of the D1/D5 CFT, JHEP 12 (2018) 050 [arXiv:1811.00155]. — A modern symmetry-oriented entry to D1–D5, useful for tensionless regimes.

[45] J. a. E. Alday, G. Giribet and S. A. Hansen, The AdS$_3$ Virasoro-Shapiro amplitude, arXiv:2402.00050. — A representative entry to the modern $AdS_3$ string amplitude program.

[46] S. M. Chester and X. Zhong, The AdS$_3$ Virasoro-Shapiro amplitude in the background of NS-NS flux, Phys. Rev. Lett. 134 (2025) 151602 [arXiv:2311.03476]. — Computations in NS–NS flux; connects worldsheet control to CFT data.

[47] J. a. E. Alday, L. Eberhardt, K. Haddad, S. M. Harrison and S. S. Tirumala, The AdS$_3\times S^3$ Virasoro-Shapiro amplitude: the matrix-element approach, arXiv:2412.06429. — CFT-data-driven approach; useful for RR flux and beyond.

[48] Y. Jiang, L. Eberhardt, S. M. Chester and B. C. van Rees, D1-D5 CFT data from the AdS$_3\times S^3$ Virasoro-Shapiro amplitude in pure RR flux, arXiv:2601.18646. — Very recent progress (pure RR flux) connecting amplitudes to D1–D5 CFT data.

17.7 Integrability in $AdS_3/CFT_2$

[49] A. Sfondrini, Towards integrability for AdS$_3$/CFT$_2$, arXiv:1406.2971. — An early review perspective; still useful for the basic landscape.

[50] A. Seibold and A. Sfondrini, AdS$_3$ Integrability, Tensionless Limits, and Deformations, arXiv:2408.08414. — A modern review; recommended entry point for the current state of the field.

[51] S. Frolov and A. Sfondrini, Massless S matrices for AdS$_3$/CFT$_2$, JHEP 04 (2022) 067 [arXiv:2112.08895]. — Massless-mode subtleties and worldsheet S-matrix constraints.

[52] S. Frolov and A. Sfondrini, Mirror Thermodynamic Bethe Ansatz for AdS$_3$/CFT$_2$, JHEP 03 (2022) 138 [arXiv:2112.08898]. — A systematic TBA treatment including massive and massless modes.

[53] S. Ekhammar, N. Gromov and B. Stefański Jr., Demystifying the Massless Sector in AdS$_3$ Quantum Spectral Curve, JHEP 10 (2025) 188 [arXiv:2412.11915]. — Recent QSC progress clarifying massless-sector solutions and dressing phases.

17.8 Solvable irrelevant deformations

[54] F. A. Smirnov and A. B. Zamolodchikov, On space of integrable quantum field theories, Nucl. Phys. B 915 (2017) 363 [arXiv:1608.05499]. — The foundational $T\bar T$ paper.

[55] L. McGough, M. Mezei and H. Verlinde, Moving the CFT into the bulk with $T\bar T$, JHEP 04 (2018) 010 [arXiv:1611.03470]. — The main cutoff-$AdS_3$ holographic proposal.

[56] A. Giveon, N. Itzhaki and D. Kutasov, TTbar and LST, JHEP 07 (2017) 122 [arXiv:1701.05576]. — Connects $T\bar T$ deformations to stringy UV completions.

[57] M. Guica, An integrable Lorentz-breaking deformation of two-dimensional CFTs, SciPost Phys. 5 (2018) 048 [arXiv:1710.08415]. — The $J\bar T$ deformation and its holographic relevance.